The model for the frog population was given by the function F(m)=m+6, and the model for the turtle population was given by the function T(m)=m^2+3/m+1. Which of the following expressions defines the function F(m)-T(m), the difference in size between the two populations m months after the initial count?

Rationale

F(m) - T(m) is defined by the expression 7m + 3/(m + 1).

To find the difference between the frog population function F(m) = m + 6 and the turtle population function T(m) = (m^2 + 3)/(m + 1), we perform the subtraction F(m) - T(m). Simplifying this expression yields 7m + 3/(m + 1), which captures the desired difference in population sizes.

A (7m+9/m+1) YOUR ANSWER

This expression incorrectly adds 3 to 6, resulting in 9 instead of the correct 3. The subtraction of T(m) from F(m) involves the difference in the constant terms after the algebraic simplification, which is miscalculated here.

B (-7m-3/m+1) YOUR ANSWER

This option suggests a negative difference between the two populations. However, the actual subtraction yields a positive result, showing that the frog population increases relative to the turtle population after performing the calculations. The signs of the terms are thus incorrect.

C (7m+3/m+1) CORRECT

This expression accurately represents the difference between the two populations after the necessary algebraic operations are performed. The coefficients and constant term have been correctly derived, reflecting the expected population difference.

D (#NAME?) YOUR ANSWER

This choice indicates a name error, suggesting that the expression is not defined or is improperly formed. It lacks any relevant mathematical meaning in the context of the problem.

E (#NAME?) YOUR ANSWER

Similar to option D, this choice also presents an undefined expression. It fails to represent a valid mathematical function, rendering it irrelevant to the original question.

Conclusion

The correct expression for the difference in population sizes between frogs and turtles after m months is 7m + 3/(m + 1). This result is derived by correctly simplifying the subtraction of the turtle population model from the frog population model. The other options either miscalculate the terms, introduce incorrect signs, or do not represent valid mathematical expressions, reinforcing the accuracy of the chosen answer.

F(m) - T(m) is defined by the expression 7m + 3/(m + 1).

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